Optical cross-connect switch

00002This application claims the benefit of Provisional Application No. 60/373,621, filed Apr. 19, 2002, the entire contents of which is hereby incorporated by reference in this application.

FIELD OF THE INVENTION

00003The present invention relates generally to optical beam-steering, spatial light modulation, diffractive optics and computer-generated holography, and more particularly to all-optical cross-connect switches which may be used in, inter alia, optical communication systems.

BACKGROUND OF THE INVENTION

00004Currently MEMS is the leading technology for implementing all-optical cross-connect switches for fiber optic telecommunication networks. Typically these devices work by collimating light from an array of input fibers and reflecting these light beams off an array of movable micromirrors, which then act to deflect each beam toward the appropriate output fiber. For large switches (more than 100 input and output fibers) the most popular configuration has the micromirrors arranged on a rectangular grid with their faces parallel to the substrate. The input light beams are directed at the micromirror array in the normal direction, and the mirrors tilt on two axes to deflect each input beam in the desired direction. Very often the deflected light beams are retro-reflected by an ordinary bulk mirror back onto other micromirrors in the array. These mirrors are then tilted so as to direct the light beams onto the appropriate fibers in the original fiber array. This way, any fiber can be cross connected to any other fiber in the array without having to partition the fibers into input and output groups and restricting cross-connections to between these groups only.

00005This so-called “3D MEMS” configuration has several drawbacks: <ul id="ul100001" list-style="none"><li id="ul100002-li00002"><ul id="ul100002" list-style="none"><li id="ul100002-p00006" num="00006">(1) Switching Speed: The mirrors are 100's of microns in diameter and fairly thick (to prevent warping). Their large size and mass give them a large moment of inertia, which limits their tilting speed to about 10 ms. Moreover this problem becomes exacerbated as the number ports in the switch increases, because this implies a smaller angular separation between ports. In turn, this means that beam spreading due to Gaussian optics and diffraction must be reduced, and only increasing the mirror diameters can do that.</li><li id="ul100002-p00007" num="00007">(2) Control: Using a 1024×10²⁴switch as an example, a switch this size would have a 32×32 fiber array, and therefore the mirror tilt would have to be controlled to a precision considerably greater than {fraction (1/32)} of full deflection. This requirement only becomes more stringent as the switch size increases.</li><li id="ul100002-p00008" num="00008">(3) Fabrication: A large mirror diameter requires that the mirrors be suspended high above the substrate so that there is sufficient clearance for the mirror to tilt to its maximum deflection (typically about 10°). These kinds of structures are difficult to fabricate with currently well-established micromachining techniques. Often, designs resort to pop-up structures that are fabricated with thin, surface micromachined structures that fold up into their final configuration. However this approach significantly complicates the device. Furthermore, making large diameter mirrors that are stiff and optically flat is difficult.</li><li id="ul100002-p00009" num="00009">(4) Actuation: Arrays of tilt-mirrors are typically actuated electrostatically. However this type of actuation has a possible collapse instability (“snap-down”), which arises because the electrostatic force varies as the inverse square of the electrode separation, whereas the mechanical restoring force typically increases only linearly. To avoid this instability, the minimum electrode separation must be at least ⅓ of the initial gap between electrode and mirror. In turn, this means that the already large clearances required by large diameter mirrors must be three times larger still. Furthermore, the mirror's large moment of inertia will require large electrostatic and restoring forces, if it is to have reasonable switching times. In turn, this means that large driving voltages (˜100V) are needed.</li></ul></li></ul>

00010The conception of tilt-mirror optical switches is based on geometric optics, i.e., light rays from the input fiber are reflected to the desired output fiber, and the appropriate tilt angle for the micromirror are calculated from the geometric optics law “the angle of incidence equals the angle of reflection.” Alternatively, designs can be based on the more general theory of wave optics, where light is understood to be an electromagnetic wave rather than a geometric ray. In this picture, an optical switch element alters the amplitude and/or phase profile of the incoming wavefront so that the light wave then propagates toward the desired output fiber. In the case of the tilt-mirror switch, the tilted mirror induces in the wavefront a phase delay that varies linearly across the face of the mirror, and it is this linearly tapered phase profile that is responsible for redirecting the light wave propagation.

00011Instead of a continuous linear taper, one might look for a different phase profile that would steer the outgoing light beam. One example is the variable-blaze diffraction grating. On the other hand, the most obvious possibility is to simply replace the continuous linear phase profile of the tilt-mirror with a staircase approximation, which is the principle on which phased array antennas work. A tiling of small mirrors that move up and down in a piston-like fashion can generate such a staircase phase profile. This array is essentially a replacement for a single tilt-mirror, so an N×N switch would thus consists of 2N copies of this array, two for each input beam. This approach has several advantages over tilt-mirrors: <ul id="ul100003" list-style="none"><li id="ul100004-li00004"><ul id="ul100004" list-style="none"><li id="ul100002-p00012" num="00012">(1) Switching Speed: The much smaller diameter of these mirrors reduces their masses in two ways. First, the area of each mirror is much smaller. Second, the smaller mirror does not need to be as thick to maintain optical flatness. This drastic reduction in mass means that the mirror can be switched much faster—in about 10 μs.</li><li id="ul100002-p00013" num="00013">(2) Control: Although these mirrors have about the same number of positions as the tilt-mirrors (16 vertical positions vs. 32 tilt-angles along a given axis), the precision with which these positions need to be controlled is much less. This scheme is more tolerant to positioning errors.</li><li id="ul100002-p00014" num="00014">(3) Reliability: The performance of this device will degrade gradually if individual mirror elements fail. In contrast, almost any failure mode of the tilt-mirror will have a catastrophic impact on its performance.</li><li id="ul100002-p00015" num="00015">(4) Fabrication: The vertical displacement of the mirrors will not need to exceed ½λ, which is 780 nm for the fiber optic communications C-band. Therefore the required clearances are much smaller, and the mirror structures can be fabricated using straightforward surface micromachining techniques.</li><li id="ul100002-p00016" num="00016">(5) Actuation: These small clearances also allow smaller driving voltages.</li></ul></li></ul>

00017Despite these advantages, the phased array approach has one very large drawback, i.e., the huge number of mirrors that must be fabricated. Because of diffraction, the reflected beam has a small spreading angle. This limits the number of non-overlapping steering angles, and this number is a function of the number mirrors in the array. A micromirror array producing a 32×32 grid of distinct steering angles will need to have at least 100×100 elements. Therefore a 1000×1000 OXC switch based on 100×100 micromirror arrays will have 20 million mirrors. Even so, an array this size will have a diffraction efficiency of only about 80%, and a 99% efficient array would need to be at least 512×512 elements.

00018Not only does this many mirrors need to be fabricated with good yield, but the control circuitry must also be duplicated 20 million times as well. Even merely addressing and running control lines to this many mirrors is a formidable challenge.

SUMMARY OF THE INVENTION

00019It is therefore an object of the present invention to reduce the size and complexity of the mirror arrays needed by free-space all-optical cross-connect switches to render these switches more practical and manufacturable.

00020It is another object of the present invention to reduce the complexity of the electronic control system for the switch.

00021It is a further object of the present invention to allow for the substitution of the micromirror array by other types of spatial light modulators to render faster switching speeds or less expensive to manufacture.

00022It is yet another object of the present invention to allow channel splitting, wherein the light signals from a single input optical fiber is split and directed to several output optical fibers.

00023It is yet a further object of the present invention to allow channel merging, wherein the light from several input optical fibers are directed to the same output optical fiber.

00024It is still a further object of the present invention to enable a considerably more manufacturable optical cross-connect switch than previous designs.

00025These objectives are realized in the present invention by using a novel and very flexible architecture for an optical cross-connect switch that allows the use of almost any spatial light modulator (SLM) within the optical cross-connect switch to effect the switching action. The SLM can be very rudimentary, i.e., and all that is necessary is that it have relatively few segments and be capable of generating phase-shifts of 0 and π. The utility the cross-connect switch of the present invention is that it is easier to realize an inexpensive, easily manufacturable or fast SLM that has few segments and only generates two levels of phase-shifts. Since the dynamic (switching) element of any optical cross-connect switch is the most expensive and most difficult to manufacture element, this advantage has a large impact on the utility of the switch.

00026For example, if a micromirror array were to be used for the SLM, then the fact that only phase-shifts of 0 and π are required means that each micromirror needs to move to two different positions. In turn, these two displacements can be set using mechanical stops, obviating the need for closed-loop electronic control. Moreover, a micromirror array of fewer mirrors can be fabricated with higher yields.

00027The present invention results from the realization that the micro-mirror array in the optical cross-connect switches described in “Background of the Invention” are required to perform two functions simultaneously: (1) state selection where a re-configuration of the mirror array changes which output fiber is selected to receive the light from a given input fiber, and (2) beam shaping in which the light is focused onto the selected output fiber to maximize output-coupling and minimize cross-coupling with other fibers. On one hand, the micromirror array needs to be dynamic to perform state selection. On the other hand, the need for good focusing (or collimation) of the output beam is the reason why so many mirrors are needed in a phased array and why tilt-mirrors need to be so wide. In the optical switch of the present invention, these two functions are separated. The micromirror array is responsible only for state selection without regard to either collimation or even steering of the reflected light; all that is important is that the array can produce a distinct, linearly independent phase profile for each output state (i.e., selected output fiber). A fixed optical subsystem is then added whose sole function is beam shaping. This beam shaper transforms each phase profile reflected by the mirror array into a wave-amplitude profile that converges onto the appropriate output fiber.

00028In this scenario, the mirror array acts as a phase-only spatial light modulator (SLM). Therefore, the mirror array can be substituted with an SLM implemented in another technology, e.g. an electro-optic or a liquid crystal.

00029An (L×K)-switch according to the present invention can be built up by ganging L (1×K) switch elements in parallel using fiber optic couplers to connect output fibers. One embodiment of a (1×K)-switch element according to the present invention consists of a lens that collimates light from a single input optical fiber so that a plane wave is incident on a spatial light modulator (SLM), such as a micromirror array. The SLM is set to imposes any one of K different complex wave-amplitude (or “wavefront”) profiles on the wave. The wave-amplitude profiles can be very irregular because subsequent beam shaping optics transforms the wave into one that focuses on to a desired one of a plurality of output optical fibers. Depending on which profile the incident wave-amplitude has, the beam shaper focuses the light onto a different output fiber Employing beam shaping optics allows great flexibility in choosing the K wavefront profiles for the K switch states. Since it is irrelevant as to what these profiles look like, as long as they are distinct, it is possible to choose a very simple, easy to manufacture and/or fast design for the SLM. There is also a constraint on the wavefront profiles required from the SLM, i.e., it must be linearly independent. In any optical system composed of linear elements, including a beam shaper, the complex amplitudes for the incoming and outgoing waves are related by the integral operator: <maths id="MATH-US-00001" num="00001"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mi>v</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>∫</mo><mrow><mo>∫</mo><mrow><mrow><mi>G</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mrow><mi>y</mi><mo>|</mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo>,</mo><msup><mi>y</mi><mi>′</mi></msup></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><mi>u</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>′</mi></msup><mo>,</mo><msup><mi>y</mi><mi>′</mi></msup></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><mo>ⅆ</mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo>⁢</mo><mrow><mo>ⅆ</mo><msup><mi>y</mi><mi>′</mi></msup></mrow></mrow></mrow></mrow><mo>≡</mo><mrow><mover><mi>G</mi><mo>^</mo></mover><mo>⁢</mo><mrow><mi>u</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where υ is the complex amplitude of the outgoing wave evaluated at the exit plane, and u is the complex amplitude of the incoming wave evaluated at the entrance plane. Considering three different wavefronts produced by the state-selecting SLM, u<i>i</i>(x,y), i=1,2,3, and the corresponding output wavefronts υ, (x,y)=Ĝu_i(x,y), and assuming that the output waves with amplitude profiles v₁and v₂converge on fibers #1 and #2 as desired, and that u₃is not linearly independent of u₁and u₂, then u₃=c₁u₁+c₂u₂. But since the integral operator is linear, it can be deduced that <br />υ₃<i>=Ĝu</i>₃<i>=Ĝ</i>(<i>c</i>₁<i>u</i>₁<i>+c</i>₂<i>u</i>₂)=<i>c</i>₁<i>Ĝu</i>₁<i>+c</i>₂<i>Ĝu</i>₂<i>=c</i>₁υ₁<i>+c</i>₂υ₂  (2)<br /> Therefore, portions of the light from output υ₃converge on both fibers #1 and #2. Thus to avoid this, all the input wavefront profiles, u_i(x,y), must be linearly independent.

00033One useful side-effect is the ability to broadcast an input signal to more than one output. For instance, if one SLM configuration focuses light on output fiber A and another configuration focuses light on fiber B, then a linear combination of these two configurations will focus light on both A and B simultaneously. Each state n of the SLM produces a different complex wave-amplitude, u_n(x,y), at its output. The beam shaper needs to transform each u_nto a complex amplitude υ_nthat will produce focusing on output fiber n. In other words, <br />υ_n(<i>x,y</i>)=<i>Ĝu</i>_n(<i>x,y</i>) <i>n=</i>1,2<i>. . . N</i>  (3)<br /> Thus, the beam shaper needs to have an amplitude transfer function G(x,y|x′,y′) that satisfies equations (1) and (3). All such transfer functions are too complex to be realized by using traditional optics. However, the inclusion of diffractive optical elements (DOEs) provides enough flexibility that (1) and (3) can be satisfied for appropriately chosen υ_n.

00036The simplicity of the SLM used in the present invention is offset by the complexity of the beam shaper in the present invention, but the fact that this system element is fixed greatly simplifies its manufacture.

BRIEF DESCRIPTION OF DRAWINGS

00037<figref idref="DRAWINGS">FIG. 1</figref> is a schematic representation of M (1×N)-switches ganged together to make an (M×N) optical switch.

00038<figref idref="DRAWINGS">FIG. 2</figref> is a layout of a (1×N)-switch using a reflective spatial light modulator (SLM) and a single diffractive optical element (DOE).

00039<figref idref="DRAWINGS">FIG. 3</figref> is a layout of a (<b>1</b>×N)-switch using a reflective SLM, a single DOE, and coupling lenses.

00040<figref idref="DRAWINGS">FIG. 4</figref> is a layout of a (1×N)-switch using a reflective SLM and multiple DOEs.

00041<figref idref="DRAWINGS">FIG. 5</figref> is a layout of a (1×N)-switch using a reflective SLM and multiple DOEs, with one DOE preceding the SLM.

00042<figref idref="DRAWINGS">FIG. 6</figref> is a layout of an (M×N)-switch using a reflective spatial light modulator (SUM) and a single diffractive optical element (DOE).

00043<figref idref="DRAWINGS">FIG. 7</figref> is a layout of a (1×N)-switch using a transparent spatial light modulator (SLM) and a single diffractive optical element (DOE).

00044<figref idref="DRAWINGS">FIG. 8</figref> is a schematic of a representative (1×K)-switch element.

00045<figref idref="DRAWINGS">FIGS. 9A and 9B</figref> show switch topologies for direct implantation and ganged (1×K)-switch elements, respectively.

00046<figref idref="DRAWINGS">FIGS. 10A and 10B</figref> show major functional units of a (1×K)-switch element, i.e., contiguous beam-shaping optics and split beam-shaping optics, respectively.

00047<figref idref="DRAWINGS">FIGS. 11A and 11B</figref> show state selectors that are a transparent SLM and a reflective SLM, respectively.

00048<figref idref="DRAWINGS">FIGS. 12A-12C</figref> show beam shapers that are a single-DOE embodiment, a double-DOE embodiment, and a n-DOE embodiment, respectively.

00049<figref idref="DRAWINGS">FIG. 13</figref> shows a direct implementation of an (L×K) switch using spatial frequency multiplexed inputs.

DETAILED DESCRIPTION OF THE INVENTION

00050The present invention is directed to optical cross-connect switches. An (M×N) optical switch <b>10</b> according to the present invention can be formed by ganging M (1×N)-switches <b>11</b> together, as depicted in FIG. <b>1</b>. These switches <b>11</b> can be connected using optical fibers <b>20</b> and combiners <b>24</b>. One embodiment for a (1×N) switch <b>11</b> is depicted in <figref idref="DRAWINGS">FIG. 2</figref>, where light (not shown) from fiber <b>14</b> is collimated by lens <b>12</b> and impinges on a reflective spatial light modulator (SLM) <b>16</b>. Upon reflection, the light passes through a diffractive optical element (DOE) <b>18</b> and is focused on the desired output coupling lens <b>19</b>. Each lens <b>19</b>, which is preferably either a micro-lens or a GRIN lens, then further focuses the light into the end of the selected output fiber <b>20</b>. The output coupling lens <b>19</b> and output fiber <b>20</b> are members of an output array <b>22</b> of N lens/fiber pairs. Depending on which output fiber is “selected”, the SLM <b>16</b> assumes a configuration that imposes a unique wavefront shape on the reflected light. The DOE <b>18</b> then transforms each of these different wavefronts such that the output is focused onto the appropriate output coupling lens <b>19</b> and fiber <b>20</b>. For some configurations of SLM <b>16</b>, the light can be focused on multiple pairs of lens <b>19</b>/fibers <b>20</b>. For instance, if one SLM configuration focuses light on output fiber <b>20</b>A and another configuration focuses light on fiber <b>20</b>B, then a linear combination of these two configurations will focus light on both fibers <b>20</b>A and <b>20</b>B simultaneously.

00051Another embodiment of a (1×N)switch <b>11</b>A is shown in FIG. <b>3</b>. Switch <b>11</b>A uses coupling lenses <b>15</b> between SLM <b>16</b> and DOE <b>18</b> and between DOE <b>18</b> and the output fiber array <b>22</b>. The spacing between the coupling lenses <b>15</b> and the other optical elements is set at the focal length of the lens <b>15</b>. In this embodiment, the light impinging on the DOE <b>18</b> is the Fraunhofer diffraction pattern of the SLM mirror array <b>16</b>, and the light impinging on the output fiber array is the Fraunhofer pattern of DOE <b>18</b>'s output. This arrangement greatly simplifies design at the expense of adding more lenses.

00052Yet another embodiment of a (1×N)switch <b>11</b>B is shown in FIG. <b>4</b>. Switch <b>11</b>B uses two or more DOEs <b>18</b> separated by free-space. At the expense of adding more DOEs <b>18</b>, this alternative embodiment has the advantage of allowing even more flexibility in the optical transfer function that can be realized in the beam-shaping system element.

00053Another embodiment of a (1×N)switch <b>11</b>C is shown in FIG. <b>5</b>. Here, a second DOE <b>18</b> is placed before SLM <b>16</b> in addition to the first DOE <b>18</b> placed after SLM <b>16</b>.

00054In the embodiment of a (1×N)switch <b>11</b>D depicted in <figref idref="DRAWINGS">FIG. 6</figref>, light from an array of M input fibers <b>14</b><i>a </i>is directed onto SLM <b>16</b> through lens <b>12</b>. This configuration allows the direct realization of an (M×N) switch, without having to gang M (1×N)-switches together, as depicted in the embodiment shown in FIG. <b>1</b>. However, this simplification comes at the cost of requiring more pixels in SLM <b>16</b> and DOE <b>18</b>.

00055In each of the embodiments shown in <figref idref="DRAWINGS">FIGS. 1-6</figref>, coupling lenses can be used between each of the optical elements to simplify design at the expense of having more lenses. In each of these same embodiments, any spatial light modulator capable of altering the phase of an impinging light wavefront can also be used. This includes, but is not limited to, micro-mirror arrays, liquid crystal SLMs and electro-optic SLMs. The SLM does not need to provide high-resolution control of the wavefront's phase. In particular, the SLM needs only to generate a ±½π relative phase-shift. This attribute is especially useful if a micro-mirror array is used as the SLM. The SLM can be reflective, as depicted in <figref idref="DRAWINGS">FIGS. 2-6</figref>, or it can be transparent, as depicted for the (1×N)switch IIE shown in FIG. <b>7</b>.

00056One embodiment of an (L×K) optical switch <b>110</b> according to the present invention is shown in FIG. <b>9</b>A. For this embodiment, L (1×K)-switch elements <b>111</b> are ganged together, as depicted in FIG. <b>9</b>B. The switches <b>111</b> are connected together using optical fibers <b>120</b> and fiber optic combiners <b>124</b>. One embodiment for a (1×K)-switch element <b>110</b> is shown in FIG. <b>8</b>. Switch element <b>110</b> consists of a lens <b>112</b> that collimates light from a single input optical fiber <b>114</b> so that a plane wave (not shown) is incident on a spatial light modulator (SLM) <b>116</b>, such as a micromirror array. This SLM <b>116</b> imposes a non-uniform complex wave-amplitude profile on the wave. This profile can be very irregular, because the subsequent beam shaping optics <b>118</b> transform this wave into one that focuses on to a desired one of a plurality of output optical fibers <b>120</b> through a plurality of output coupling lens <b>119</b>. Each lens <b>119</b>, which is preferably either a micro-lens or a GRIN lens, further focuses the light into the end of the selected output fiber <b>120</b>. The SLM <b>116</b> can be set to produce any one of K different complex wave-amplitude (or “wavefront”) profiles, and the beam shaping optics <b>118</b> transform each of these input profiles into a wave that converges onto a different output fiber <b>120</b> through a corresponding lens <b>119</b>.

00057Another embodiment for the (1×K) switch element <b>111</b>A is depicted in <figref idref="DRAWINGS">FIG. 10A</figref>, where light from fiber <b>114</b> is collimated by lens <b>112</b> and impinges on a state selector <b>116</b>. Upon exiting the state selector <b>116</b>, the light wave propagates through free-space <b>117</b> to a beam shaper <b>118</b>. The light wave exiting from state selector <b>116</b> is not necessarily collimated, nor would it necessarily converge on the appropriate output fiber <b>120</b>. The beam shaping optics <b>118</b> transform the wavefront incident on its input plane into one that converges on the collection lens <b>119</b> for the appropriate output optical fiber <b>120</b>. To switch the light to a different output fiber, state selector <b>116</b> changes configuration, altering the complex wave-amplitude of the light wave exiting it. In turn, the beam shaper <b>118</b> transforms this different “wavefront” into one that converges on a different output fiber/lens pair <b>119</b>/<b>120</b>. The beam shaper <b>118</b> is a fixed optical system.

00058An embodiment of state selector <b>116</b> is a simple transparent spatial light modulator (SLM) <b>122</b>, shown in FIG. <b>11</b>A. The input plane <b>121</b> and output plane <b>123</b> of state selector <b>116</b> is, therefore, simply the front and back surfaces of the SLM <b>122</b>.

00059An alternative embodiment of the state selector <b>116</b> is a reflective SLM <b>124</b> shown in FIG. <b>11</b>B. Surfaces <b>125</b> and <b>126</b> are the input and output planes, respectively of reflective SLM <b>124</b>.

00060A principal cost and manufacturing hurdle to any optical switch is its dynamic element(s). Accordingly, a central objective of the switch architecture of the present invention is to allow flexibility in the selection of the SLM, including the use of a very basic SLM. The basic SLM produces piecewise constant amplitude-phase profiles when illuminated by a plane wave. Each SLM element produces a constant amplitude-attenuation and phase-shift across its face. For the SLM to be as simple as possible, it should have as few of these elements as possible. However, for all K of these piecewise constant profiles to be linearly independent, the SLM must have at least K elements.

00061The SLM can be further simplified if each element takes on as few states as possible. Ideally, only two states per element would be desirable for the SLM. In the case of a micromirror array SLM, the control problem for each mirror would be greatly simplified. When only two states are required, the two positions of each micromirror can be set by mechanical stops, obviating the need for active electronic control.

00062The overall optical switch should also be low loss. This implies that the SLM should not absorb any light, and therefore, the SLM should impose a phase-only profile. In sum, for the simplest lossless SLM, the wavefront profiles it produces must: <ul id="ul200001" list-style="none"><li id="ul200002-li00002"><ul id="ul200002" list-style="none"><li id="ul200002-p00063" num="00063">(i) be linearly independent;</li><li id="ul200002-p00064" num="00064">(ii) consist of K piecewise constant segments of equal size; and</li><li id="ul200002-p00065" num="00065">(iii) satisfy |u_i(x,y)|=1.</li></ul></li></ul>

00066There is an orthonormal set of functions that have these properties: the 2D Hadamard functions <maths id="MATH-US-00002" num="00002"><math overflow="scroll"><mrow><msubsup><mrow><mo>{</mo><mrow><msub><mi>had</mi><mi>mn</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>}</mo></mrow><mrow><mi>mn</mi><mo>,</mo><mrow><mo>=</mo><mn>0</mn></mrow></mrow><mi>∞</mi></msubsup><mo>.</mo></mrow></math></maths><br /> These functions are also called Walsh functions. The function had_mn(x,y) can be built up from a tiling of 2^p×2^qrectangular segments of equal size and with constant values ±1, if m≦2^pand n≦2^q, where p,q=1, 2, 3 . . .

00068Consequently, the SLM should be capable of generating ±½π phase-shifts without any absorption and have M×N elements (where MN≧K, M=2^p, N=2^q, and p,q=1, 2, 3 . . . ). Between the two embodiments for the state selector <b>116</b> depicted in <figref idref="DRAWINGS">FIGS. 10A and 10B</figref>, any phase-only SLM will work, including micromirror arrays, liquid crystal SLMs and electro-optic SLMs. Any of these embodiments for state selector <b>116</b> will produce at its exit a complex wave-amplitude from the set {had_mn(x,y)|m=1, 2, . . . M−1, n=1,2, . . . N−1}.

00069Beam shaper <b>118</b> then converts this wave into one that converges on the output fiber <b>120</b> at position (m,n). Moreover, the same fixed optical train needs to do this for all m=0,1, . . . M−1 and n=0,1, . . . N−1. There are a number of standard coherent optical processing techniques that can be adapted to perform this function. By and large, these techniques are implemented by a train of lenses and diffractive optical elements (DOE) (sometimes also called computer generated holograms (CGH), holographic masks or holographic elements). Consequently, the beam shaper <b>118</b> is laid out as depicted in <figref idref="DRAWINGS">FIGS. 12A-12C</figref>. The embodiment in <figref idref="DRAWINGS">FIG. 12C</figref>, beam shaper <b>118</b>C, consists of an optional input coupling lens <b>201</b> followed by a first DOE <b>202</b>, an optional coupling lens <b>203</b>, a second DOE <b>204</b>, another optional coupling lens <b>205</b>, etc. up to a P^thDOE <b>206</b> and, lastly, an optional output coupling lens <b>207</b>. The embodiments in <figref idref="DRAWINGS">FIGS. 12A and 12B</figref> of beam shapers <b>118</b>A and <b>118</b>B, respectively, are special cases containing exactly one or two DOEs, respectively. The coupling lenses between DOEs (e.g., <b>202</b>,<b>204</b>) are Fourier transforming lenses. Therefore, the preceding and succeeding DOEs are located in the front and back focal planes of each lens. These lenses can greatly simplify design at the expense of adding more lenses. The input coupling lens <b>201</b> and the output coupling lens <b>207</b> can be either Fourier transforming or imaging lenses.

00070In general, these diffractive optical elements can have a complex transfer function. However, practical DOEs usually have either a pure-amplitude (i.e., real) or a pure-phase transfer function. Nevertheless, these simpler DOEs can be arranged to have an effective complex transfer function by utilizing any one of a number of standard “encoding” techniques. Amplitude-only masks are generally inefficient, which is detrimental to the present switch application where as little loss as possible is desirable. On the other hand, systems employing phase-only masks, often called “kinoforms”, can theoretically have up to 100% efficiency, since the mask's complex transfer function (CTF) satisfies |h(x,y)|=1, and is absorptionless by definition.

00071Therefore, phase-only DOEs can be used in the present invention, and they are available in several varieties. The preferred type is the surface-relief DOE which, in turn, comes in two varieties: the continuous (or “analog”) relief DOE and the multilevel (or “binary optics”) DOE. Surface-relief DOEs are fabricated by any one of several microlithographic techniques, and they can be custom manufactured by companies such as MEMS Optical of Huntsville, Ala. or Wavefront Sciences of Albuquerque, N. Mex.

00072In an embodiment of the present invention, a system capable of performing a general spectral analysis (i.e., a general “optical integral transform”) is adapted to function as the beam shaper <b>118</b>. In a traditional general spectral analyzer, the complex wave-amplitude incident on its input plane is expanded in a generalized Fourier series of some desired set of orthonormal basis functions, <maths id="MATH-US-00003" num="00003"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mi>u</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>m</mi><mo>,</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow></mrow><mi>∞</mi></munderover><mo>⁢</mo><mrow><msub><mi>u</mi><mi>mn</mi></msub><mo>⁢</mo><mrow><msub><mi>ψ</mi><mi>mn</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> and light intensities proportional to u_mnare focused onto K photodiodes arrayed in the analyzer's output plane. This way, the first K Fourier coefficients are measured. Such a device can be adapted to function as the beam-shaping subsystem <b>118</b> needed for the optical switch <b>110</b>. In this embodiment, the basis functions are chosen to be 2D Hadamard functions, the output of the state selector <b>116</b> is imaged onto the analyzer's input plane, and the photodetectors are replaced by the collection lens/output fiber pairs <b>119</b> and <b>120</b>. Then, when the state selector <b>116</b> is set to produce had_pq(x,y), for example, all of the light will be directed to the collection lens/output fiber pair <b>119</b> and <b>120</b> sitting at the location for u_pq.

00074Several implementations of a Hadamard spectral analyzer, or “optical Walsh-Hadamard transform”, have been previously developed. A methodology for designing systems that perform general optical transforms has also been developed by Gu, et al. See B. Gu, G. Yang, and B. Dong, Appl. Opt., 25, 3197 (1986). The entire contents of the article by Gu, et al. is hereby incorporated by reference in this application. These systems consist of a train of diffractive optical elements, and this method works for an arbitrary number of DOEs with or without coupling lenses. Thus, these systems of Gu, et al. are applicable to the embodiment of a beam shape depicted in FIG. <b>12</b>C. For a general system composed of P diffractive optical elements, the input/output relation linking the complex wave-amplitude on the input plane to the complex amplitude on the output plane is given by the integral operator <maths id="MATH-US-00004" num="00004"><math overflow="scroll"><mtable><mtr><mtd><mtable><mtr><mtd><mrow><mrow><mi>v</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mi /><mo>⁢</mo><mrow><mo>∫</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mrow><mi>⋯</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mrow><mo>∫</mo><mrow><mrow><msub><mi>G</mi><mi>P</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mrow><mi>y</mi><mo>|</mo><msub><mi>x</mi><mi>P</mi></msub></mrow><mo>,</mo><msub><mi>y</mi><mi>P</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><msub><mi>H</mi><mi>P</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>P</mi></msub><mo>,</mo><msub><mi>y</mi><mi>P</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><msub><mi>G</mi><mrow><mi>P</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>P</mi></msub><mo>,</mo><mrow><msub><mi>y</mi><mi>P</mi></msub><mo>|</mo><msub><mi>x</mi><mrow><mi>P</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>,</mo><msub><mi>y</mi><mrow><mi>P</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>⋯</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mrow><msub><mi>H</mi><mn>1</mn></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>×</mo></mrow></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mi /><mo>⁢</mo><mrow><mrow><msub><mi>G</mi><mn>0</mn></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>|</mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo>,</mo><msup><mi>y</mi><mi>′</mi></msup></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><mi>u</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>′</mi></msup><mo>,</mo><msup><mi>y</mi><mi>′</mi></msup></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><mo>ⅆ</mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo>⁢</mo><mrow><mo>ⅆ</mo><msup><mi>y</mi><mi>′</mi></msup></mrow><mo>⁢</mo><mrow><mo>ⅆ</mo><msub><mi>x</mi><mn>1</mn></msub></mrow><mo>⁢</mo><mrow><mo>ⅆ</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>⋯</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mrow><mo>ⅆ</mo><msub><mi>x</mi><mi>P</mi></msub></mrow><mo>⁢</mo><mrow><mo>ⅆ</mo><msub><mi>y</mi><mi>P</mi></msub></mrow></mrow></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where υ(x,y) is the complex amplitude on the output plane, u(x,y) is the complex amplitude on the input plane, H_n(x_my_n) is the complex transfer function of the n^thDOE, and G_n(x_my_n|x_n−1, y₋₁) is the propagation kernel for wave propagation from the n^thto the n+1^thDOE. G₀is the kernel for propagation from the input plane to the 1^stDOE, and G_Pis the propagation kernel from the P^thDOE to the output plane. If there is only free-space separating DOEs n and n+1, then G_nis a modified Fresnel kernel <maths id="MATH-US-00005" num="00005"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>G</mi><mi>n</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mrow><mi>y</mi><mo>|</mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo>,</mo><msup><mi>y</mi><mi>′</mi></msup></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>d</mi><mi>n</mi></msub></mrow></mfrac><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mn>2</mn><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>π</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mfrac><msub><mi>d</mi><mi>n</mi></msub><mi>λ</mi></mfrac></mrow></msup><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mrow><mfrac><mi>π</mi><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>d</mi><mi>n</mi></msub></mrow></mfrac><mo>[</mo><mrow><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>-</mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mrow><mi>y</mi><mo>-</mo><msup><mi>y</mi><mi>′</mi></msup></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow></mrow></msup></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where d_nis the distance between the DOEs, and λ is the light wavelength. On the other hand, if a Fourier coupling lens is used, then G_nis a modified Fourier kernel <maths id="MATH-US-00006" num="00006"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>G</mi><mi>n</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mrow><mi>y</mi><mo>|</mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo>,</mo><msup><mi>y</mi><mi>′</mi></msup></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow></mfrac><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mn>4</mn><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>π</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mfrac><mi>f</mi><mi>λ</mi></mfrac></mrow></msup><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mrow><mo>-</mo><mi>ⅈ</mi></mrow><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><mi>π</mi></mrow><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><msup><mi>xx</mi><mi>′</mi></msup><mo>+</mo><msup><mi>yy</mi><mi>′</mi></msup></mrow><mo>)</mo></mrow></mrow></msup></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where f is the focal length of the lens. Since only the intensity |υ(x,y)|²on the output plane is of interest, the constant phase factors in equations (6) and (7) have no effect, and can be ignored. In <br /> practice, all optical signals are band-limited. Therefore the complex wave-amplitudes v(x,y) and υ(x,y) can be completely represented by a suitably dense sampling in accordance with Shannon's sampling theorem. Thus the integral equation (5) is well-approximated by its discrete version: <maths id="MATH-US-00007" num="00007"><math overflow="scroll"><mtable><mtr><mtd><mrow><mtable><mtr><mtd><mrow><mrow><mi>v</mi><mo>⁡</mo><mrow><mo>[</mo><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mo>]</mo></mrow></mrow><mo>=</mo><mi /><mo>⁢</mo><mrow><munder><mo>∑</mo><mrow><mrow><mo></mo><msub><mi>m</mi><mi>P</mi></msub><mo></mo></mrow><mo>≤</mo><msub><mi>M</mi><mi>P</mi></msub></mrow></munder><mo>⁢</mo><mrow><munder><mo>∑</mo><mrow><mrow><mo></mo><msub><mi>n</mi><mi>P</mi></msub><mo></mo></mrow><mo>≤</mo><msub><mi>N</mi><mi>P</mi></msub></mrow></munder><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mrow><mi>⋯</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mrow><munder><mo>∑</mo><mrow><mrow><mo></mo><msub><mi>m</mi><mn>1</mn></msub><mo></mo></mrow><mo>≤</mo><msub><mi>M</mi><mn>1</mn></msub></mrow></munder><mo>⁢</mo><mrow><munder><mo>∑</mo><mrow><mrow><mo></mo><msub><mi>n</mi><mn>1</mn></msub><mo></mo></mrow><mo>≤</mo><msub><mi>N</mi><mn>1</mn></msub></mrow></munder><mo>⁢</mo><mrow><munder><mo>∑</mo><mrow><mrow><mo></mo><mi>m</mi><mo></mo></mrow><mo>≤</mo><msub><mi>M</mi><mn>0</mn></msub></mrow></munder><mo>⁢</mo><mrow><munder><mo>∑</mo><mrow><mrow><mo></mo><mi>n</mi><mo></mo></mrow><mo>≤</mo><msub><mi>N</mi><mn>0</mn></msub></mrow></munder><mo>⁢</mo><mrow><mrow><msub><mi>G</mi><mi>P</mi></msub><mo>⁡</mo><mrow><mo>[</mo><mrow><mi>m</mi><mo>,</mo><mrow><mi>n</mi><mo>|</mo><msub><mi>m</mi><mi>P</mi></msub></mrow><mo>,</mo><msub><mi>n</mi><mi>P</mi></msub></mrow><mo>]</mo></mrow></mrow><mo>⁢</mo><mrow><msub><mi>H</mi><mi>P</mi></msub><mo>⁡</mo><mrow><mo>[</mo><mrow><msub><mi>m</mi><mi>P</mi></msub><mo>,</mo><msub><mi>n</mi><mi>P</mi></msub></mrow><mo>]</mo></mrow></mrow><mo>⁢</mo><mrow><msub><mi>G</mi><mrow><mi>P</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>⁡</mo><mrow><mo>[</mo><mrow><msub><mi>m</mi><mi>P</mi></msub><mo>,</mo><mrow><msub><mi>n</mi><mi>P</mi></msub><mo>|</mo><msub><mi>m</mi><mrow><mi>P</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>,</mo><msub><mi>n</mi><mrow><mi>P</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>]</mo></mrow></mrow><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>⋯</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mrow><msub><mi>H</mi><mn>1</mn></msub><mo>⁡</mo><mrow><mo>[</mo><mrow><msub><mi>m</mi><mn>1</mn></msub><mo>,</mo><msub><mi>n</mi><mn>1</mn></msub></mrow><mo>]</mo></mrow></mrow><mo>×</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mi /><mo>⁢</mo><mrow><mrow><msub><mi>G</mi><mn>0</mn></msub><mo>⁡</mo><mrow><mo>[</mo><mrow><msub><mi>m</mi><mn>1</mn></msub><mo>,</mo><mrow><msub><mi>n</mi><mn>1</mn></msub><mo>|</mo><msup><mi>m</mi><mi>′</mi></msup></mrow><mo>,</mo><msup><mi>n</mi><mi>′</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>⁢</mo><mrow><mi>u</mi><mo>⁡</mo><mrow><mo>[</mo><mrow><msup><mi>m</mi><mi>′</mi></msup><mo>,</mo><msup><mi>n</mi><mi>′</mi></msup></mrow><mo>]</mo></mrow></mrow></mrow></mrow></mtd></mtr></mtable><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><mrow><msub><mi>H</mi><mi>k</mi></msub><mo>⁡</mo><mrow><mo>[</mo><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mo>]</mo></mrow></mrow><mo>=</mo><mrow><msub><mi>H</mi><mi>k</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><mi>m</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>x</mi><mi>k</mi></msub></mrow><mo>,</mo><mrow><mi>n</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>y</mi><mi>k</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><mrow><mi>u</mi><mo>⁡</mo><mrow><mo>[</mo><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mo>]</mo></mrow></mrow><mo>=</mo><mrow><mi>u</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><mi>m</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>x</mi><mn>0</mn></msub></mrow><mo>,</mo><mrow><mi>n</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>y</mi><mn>0</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><mrow><mi>v</mi><mo>⁡</mo><mrow><mo>[</mo><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mo>]</mo></mrow></mrow><mo>=</mo><mrow><mi>v</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><mi>m</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>x</mi></mrow><mo>,</mo><mrow><mi>n</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> Where DOEs k and k+1 are separated by free-space, <maths id="MATH-US-00008" num="00008"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>G</mi><mi>k</mi></msub><mo>⁡</mo><mrow><mo>[</mo><mrow><mi>m</mi><mo>,</mo><mrow><mi>n</mi><mo>|</mo><msup><mi>m</mi><mi>′</mi></msup></mrow><mo>,</mo><msup><mi>n</mi><mi>′</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>=</mo><mi /><mo>⁢</mo><mrow><mi>exp</mi><mo>[</mo><mrow><mfrac><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>π</mi></mrow><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>d</mi><mi>k</mi></msub></mrow></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msubsup><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup></mrow><mo>+</mo><mrow><msup><mi>m</mi><mrow><mi>′</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mn>2</mn></mrow></msup><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msubsup><mi>x</mi><mi>k</mi><mn>2</mn></msubsup></mrow><mo>-</mo><mrow><mn>2</mn><mo>⁢</mo><mi>m</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msup><mi>m</mi><mi>′</mi></msup><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>x</mi><mi>k</mi></msub></mrow><mo>+</mo></mrow></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mi /><mo>⁢</mo><mrow><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msubsup><mi>y</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup></mrow><mo>+</mo><mrow><msup><mi>n</mi><mrow><mi>′</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mn>2</mn></mrow></msup><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msubsup><mi>y</mi><mi>k</mi><mn>2</mn></msubsup></mrow><mo>-</mo><mrow><mn>2</mn><mo>⁢</mo><msup><mi>nn</mi><mi>′</mi></msup><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>y</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>y</mi><mi>k</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>]</mo></mrow><mo>×</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi /><mo>⁢</mo><mrow><mrow><mrow><mi>sinc</mi><mo>[</mo><mrow><mfrac><msub><mi>w</mi><mi>k</mi></msub><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>d</mi><mi>k</mi></msub></mrow></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><mi>m</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>-</mo><mrow><msup><mi>m</mi><mi>′</mi></msup><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>x</mi><mi>k</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mo>⁢</mo><mrow><mi>sinc</mi><mo>[</mo><mrow><mfrac><msub><mi>w</mi><mi>k</mi></msub><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>d</mi><mi>k</mi></msub></mrow></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><mi>n</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>y</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>-</mo><mrow><msup><mi>n</mi><mi>′</mi></msup><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>y</mi><mi>k</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mo>,</mo></mrow></mrow></mtd></mtr></mtable></math></maths><br /> and where DOEs k and k+1 are coupled by a Fourier transforming lens, <maths id="MATH-US-00009" num="00009"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>G</mi><mi>k</mi></msub><mo>⁡</mo><mrow><mo>[</mo><mrow><mi>m</mi><mo>,</mo><mrow><mi>n</mi><mo>|</mo><msup><mi>m</mi><mi>′</mi></msup></mrow><mo>,</mo><msup><mi>n</mi><mi>′</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>=</mo><mi /><mo>⁢</mo><mrow><mi>exp</mi><mo>[</mo><mrow><mrow><mo>-</mo><mi>ⅈ</mi></mrow><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>π</mi></mrow><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><mi>m</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msup><mi>m</mi><mi>′</mi></msup><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>x</mi><mi>k</mi></msub></mrow><mo>+</mo></mrow></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mrow><mi /><mo>⁢</mo><mrow><msup><mi>nn</mi><mi>′</mi></msup><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>y</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>y</mi><mi>k</mi></msub></mrow><mo>)</mo></mrow><mo>]</mo></mrow><mo>⁢</mo><mrow><mi>sinc</mi><mo>[</mo><mfrac><mrow><mi>m</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⁢</mo><msub><mi>w</mi><mi>k</mi></msub></mrow><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow></mfrac><mo>]</mo></mrow><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mrow><mi>sinc</mi><mo>[</mo><mfrac><mrow><mi>n</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>y</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⁢</mo><msub><mi>w</mi><mi>k</mi></msub></mrow><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow></mfrac><mo>]</mo></mrow></mrow><mo>,</mo></mrow></mtd></mtr></mtable></math></maths><br /> (x_k, y_k) is the sample spacing (in the x- and y-directions) used at the k^thDOE, (x₀, y₀) is the sample spacing used at the input plane, (x, y) is the sample spacing used at the output plane, w_kis the aperture width of the k^thDOE, and w₀is the input aperture width. Because the DOEs and input have finite apertures, H_k[m,n]=0 for all |m|>M_k, |n|>N_kand u[m,n]=0 for all |m|>M₀, |n|>N₀. Therefore, the sums in equation (8) are finite, and (8) represents a finite set of simultaneous equations. Gu, et. al. disclose an iterative algorithm for solving (8) for H_k[m,n], which specify the CTFs for the diffractive optical elements.

00082Chen, et al. have used Gu, et al.'s technique to design and construct a system for performing an optical Walsh-Hadamard transform capable of generating the generalized Fourier coefficients corresponding to the first 1024 2D Hadamard functions, <maths id="MATH-US-00010" num="00010"><math overflow="scroll"><mrow><msubsup><mrow><mo>{</mo><mrow><msub><mi>had</mi><mi>mn</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>}</mo></mrow><mrow><mi>m</mi><mo>,</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow></mrow><mn>31</mn></msubsup><mo>.</mo></mrow></math></maths><br /> See Y Chen, S. Zheng, D. Li, and G. Yang, Chin. Phys. Lett., 7, 437 (1990); G. Yang, Y. Chen, S. Zheng, B. Dong, and D. Li, “A Coherent System for Performing an Optical Transform” in <i>Proceedings of the Twenty</i>-<i>Second Annual Hawaii International Conference on System Sciences, </i>v. 1, pp.445-449, IEEE Comput. Soc. Press, Washington, D.C., 1990; Y. Chen, S. Zheng, B. Dong, D. Li, and G. Yang, Appl. Opt., 27, 2608 (1988). The entire contents of the Chen, et al. articles is hereby incorporated by reference in this application. The spectral analyzer in this system consists of a single DOE plus two Fourier coupling lenses that couple this element to the input and output planes. Therefore when incorporated into the optical switch architecture of the present invention, the embodiment for the beam shaper <b>118</b> will be beam shaper <b>118</b>A shown in FIG. <b>12</b>A and will work for a 1×1024 switch element <b>10</b> (M=N=32). Both the input coupling lens <b>201</b> and the output coupling lens <b>207</b> are Fourier transforming lenses. For use in the switch of the present invention, the DOE is instead implemented by phase-encoding the CTF into a kinoform, as discussed earlier.

00084An alternative optical Walsh-Hadamard transform system previously developed is composed of a random phase mask at the input plane followed, in sequence, by a Fourier coupling lens, a second DOE, and a second Fourier coupling lens to the output plane. See J. R. Leeger and S. H. Lee, Opt. Eng., 18, 518 (1979). The entire contents of the Leeger and Lee article is hereby incorporated by reference in this application. The CTF of the first mask is <br /><i>H</i>₁(<i>x,y</i>)=<i>e</i>^iφ^{<sub2>R</sub2>}^(x,y)  (9)<br /> where φ_R(x,y) is a random, but fixed and known, function ranging from 0 to 2π. The second DOE has a CTF <maths id="MATH-US-00011" num="00011"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>H</mi><mn>2</mn></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mover><mi>h</mi><mo>^</mo></mover><mn>2</mn></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mfrac><mi>x</mi><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow></mfrac><mo>,</mo><mfrac><mi>y</mi><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>10</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where f is the focal length of the coupling lenses and ĥ₂is the Fourier transform of h₂(x,y). In turn, <maths id="MATH-US-00012" num="00012"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>h</mi><mn>2</mn></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>m</mi><mo>,</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><mo>⁢</mo><mrow><mrow><msub><mi>had</mi><mi>mn</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><mrow><mo>-</mo><mi>x</mi></mrow><mo>-</mo><mi>ma</mi></mrow><mo>,</mo><mrow><mrow><mo>-</mo><mi>y</mi></mrow><mo>-</mo><mi>na</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mo>-</mo><mrow><msub><mi>ϕ</mi><mi>R</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><mrow><mo>-</mo><mi>x</mi></mrow><mo>-</mo><mi>ma</mi></mrow><mo>,</mo><mrow><mrow><mo>-</mo><mi>y</mi></mrow><mo>-</mo><mi>na</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></msup></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>11</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where a is the spacing between photodetectors in the output plane. In the switch of the present invention, this is the spacing of the output fibers. The output of the system is then <br />υ(<i>x,y</i>)=<i>h</i>₂(<i>x,y</i>){circle around (×)}<i>u</i>(<i>x,y</i>)<i>e</i>^iφ^{<sub2>R</sub2>}⁽<i>x,y</i>)  (12)<br /> where {circle around (×)} is the correlation operator. In this alternative transform system, M=N=8. Using this system in the switch of the present invention results in a 1×64 switch element <b>10</b>, and the beam shaper <b>118</b> is the embodiment of <b>118</b>B shown in <figref idref="DRAWINGS">FIG. 12B</figref> with the addition an input coupling lens <b>201</b> that images the exit plane <b>123</b> of the state selector <b>116</b> onto the random phase DOE <b>202</b>. Alternatively, the optional input coupling lens <b>201</b> can be eliminated if the random phase DOE <b>202</b> is placed in the exit plane <b>123</b> of the state selector <b>116</b>. The output coupling lens <b>207</b> would be a Fourier transforming lens. Here again, the switch of the present invention requires that the H₁and H₂CTFs be implemented as phase-encoded kinoforms instead.

00091The beam shaper of the present invention can also be implemented using a “modan”. There are several types of modans, and one variety developed by Soifer and Golub is a DOE that, upon illumination by a plane wave, generates a superposition of free-space modes propagating in different directions. See V. Soifer and M. Golub, <i>Laser beam Mode selection by Computer Generated Holograms, </i>CRC Press, Boca Raton, Fla., 1994. The entire contents of the Soifer and Golub article is hereby incorporated by reference in this application. The complex transfer function of such an optical element is: <maths id="MATH-US-00013" num="00013"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mi>T</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover><mo>⁢</mo><mrow><msub><mi>A</mi><mi>mn</mi></msub><mo>⁢</mo><mrow><msub><mi>ψ</mi><mi>mn</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><msub><mi>α</mi><mi>mn</mi></msub><mo>⁢</mo><mi>x</mi></mrow><mo>+</mo><mrow><msub><mi>β</mi><mi>mn</mi></msub><mo>⁢</mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></msup></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>13</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where ψ_mn(x,y) is the complex amplitude of mode (m,n) evaluated at the exit plane of the modan, |A_mn| gives its relative weight, and (α_mn,β_mn) specifies the angle of propagation. Assuming that such a device is illuminated by complex amplitude ψ_pq(x,y) instead of a plane wave, and that the modan is followed by a Fourier transforming lens, then an auto-correlation peak for ψ_pqwould occur in the lens' focal plane at position <maths id="MATH-US-00014" num="00014"><math overflow="scroll"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mfrac><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow><mrow><mn>2</mn><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>π</mi></mrow></mfrac><mo>⁢</mo><msub><mi>α</mi><mi>mn</mi></msub></mrow><mo>,</mo><mrow><mfrac><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow><mrow><mn>2</mn><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>π</mi></mrow></mfrac><mo>⁢</mo><msub><mi>β</mi><mi>mn</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow></math></maths><br /> and with peak intensity |A_pq|², where f is the focal length of the lens. This arrangement is equivalent to an optical pattern recognition system based on multiplexed matched spatial filters.

00094Where a modan is used as the beam shaper <b>118</b> for a 1×K switch element <b>10</b>, 2D Hadamard functions are used as the basis modes: ψ_mn=had_mn(x,y). These modes are given equal weight, A_mn=1, and the complex transfer function is <maths id="MATH-US-00015" num="00015"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mi>T</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover><mo>⁢</mo><mrow><mrow><msub><mi>had</mi><mi>mn</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><msub><mi>α</mi><mi>mn</mi></msub><mo>⁢</mo><mi>x</mi></mrow><mo>+</mo><mrow><msub><mi>β</mi><mi>mn</mi></msub><mo>⁢</mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></msup></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>14</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> A modan with this CTF is used for the DOE <b>202</b> in the beam shaper embodiment depicted in FIG. <b>12</b>A. The input coupling lens <b>201</b> images the exit plane <b>123</b> of the state selector <b>116</b> on to the modan <b>202</b>. Therefore, the complex-amplitude incident on the modan is a 2D Hadamard function. The output coupling lens <b>207</b> is a Fourier transforming lens. As discussed in the previous paragraph, the action of the modan <b>202</b> plus the succeeding Fourier transforming lens <b>207</b> is to create an autocorrelation intensity peak in the output coupling lens' focal plane <b>208</b> (FIG. <b>12</b>A). If the state selector <b>116</b> is set to produce complex amplitude had_pq(x,y), then in the focal plane <b>208</b>, the light is concentrated around position <maths id="MATH-US-00016" num="00016"><math overflow="scroll"><mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mfrac><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow><mrow><mn>2</mn><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>π</mi></mrow></mfrac><mo>⁢</mo><msub><mi>α</mi><mi>mn</mi></msub></mrow><mo>,</mo><mrow><mfrac><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow><mrow><mn>2</mn><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>π</mi></mrow></mfrac><mo>⁢</mo><msub><mi>β</mi><mi>mn</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mrow></math></maths><br /> where a collection lens <b>19</b> and output fiber <b>20</b> are placed. The collection lens <b>19</b> acts to couple the light to the output fiber <b>20</b>. Other lens/fiber pairs are placed at locations <maths id="MATH-US-00017" num="00017"><math overflow="scroll"><mrow><mrow><mo>(</mo><mrow><mrow><mfrac><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow><mrow><mn>2</mn><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>π</mi></mrow></mfrac><mo>⁢</mo><msub><mi>α</mi><mi>mn</mi></msub></mrow><mo>,</mo><mrow><mfrac><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow><mrow><mn>2</mn><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>π</mi></mrow></mfrac><mo>⁢</mo><msub><mi>β</mi><mi>mn</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math></maths><br /> m=1,2, . . . M n=1,2, . . . N in the focal plane. Light is directed to the fiber at position <maths id="MATH-US-00018" num="00018"><math overflow="scroll"><mrow><mo>(</mo><mrow><mrow><mfrac><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow><mrow><mn>2</mn><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>π</mi></mrow></mfrac><mo>⁢</mo><msub><mi>α</mi><mi>mn</mi></msub></mrow><mo>,</mo><mrow><mfrac><mrow><mi>λ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>f</mi></mrow><mrow><mn>2</mn><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>π</mi></mrow></mfrac><mo>⁢</mo><msub><mi>β</mi><mi>mn</mi></msub></mrow></mrow><mo>)</mo></mrow></math></maths><br /> simply by setting the SLM to produce complex amplitude had_mn(x,y). To realize a DOE with CTF shown in equation (14), the DOE should be phase-only to maximize efficiency. On the other hand, CTF (14) is an amplitude-phase profile, so this CTF must be phase-encoded by any one of several methods. One method starts by recognizing that the phase of each term has no effect on the intensity profile in the focal plane <b>208</b>. Since only the intensity in the focal plane is of interest, a modan with CTF <maths id="MATH-US-00019" num="00019"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mi>T</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover><mo>⁢</mo><mrow><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>ξ</mi><mi>mn</mi></msub></mrow></msup><mo>⁢</mo><mrow><msub><mi>had</mi><mi>mn</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><msub><mi>α</mi><mi>mn</mi></msub><mo>⁢</mo><mi>x</mi></mrow><mo>+</mo><mrow><msub><mi>β</mi><mi>mn</mi></msub><mo>⁢</mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></msup></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>15</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> can be used. Therefore, ξ_mncan be manipulated as free parameters in equation (15), such that T(x,y) becomes phase-only. In other words, ξ_mnand φ(x,y) must be found, such that <maths id="MATH-US-00020" num="00020"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mi>T</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mrow><mi>ϕ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></mrow></msup><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover><mo>⁢</mo><mrow><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>ξ</mi><mi>mn</mi></msub></mrow></msup><mo>⁢</mo><mrow><msub><mi>had</mi><mi>mn</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><msub><mi>α</mi><mi>mn</mi></msub><mo>⁢</mo><mi>x</mi></mrow><mo>+</mo><mrow><msub><mi>β</mi><mi>mn</mi></msub><mo>⁢</mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></msup></mrow></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>16</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> An approximate solution to this problem can be found if α_mand β_nare chosen large enough, such that the overlap integrals <maths id="MATH-US-00021" num="00021"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>I</mi><mrow><mi>mn</mi><mo>,</mo><mi>pq</mi></mrow></msub><mo>=</mo><mrow><mo>∫</mo><mrow><mo>∫</mo><mrow><mrow><msub><mi>had</mi><mi>mn</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><msub><mi>had</mi><mi>pq</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁡</mo><mrow><mo>[</mo><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>α</mi><mi>mn</mi></msub><mo>-</mo><msub><mi>α</mi><mi>pq</mi></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><mi>x</mi></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>β</mi><mi>mn</mi></msub><mo>-</mo><msub><mi>β</mi><mi>pq</mi></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><mi>y</mi></mrow></mrow><mo>]</mo></mrow></mrow></msup><mo>⁢</mo><mrow><mo>ⅆ</mo><mi>x</mi></mrow><mo>⁢</mo><mrow><mo>ⅆ</mo><mi>y</mi></mrow></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>17</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> are small. In this case, the functions had_mn(x,y) e^i(α^{<sub2>mn</sub2>}^x+β^{<sub2>mn</sub2>}^y)are approximately orthogonal, and equation (16) can be decomposed into the simultaneous equations <maths id="MATH-US-00022" num="00022"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mtable><mtr><mtd><mrow><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>ξ</mi><mi>mn</mi></msub></mrow></msup><mo>=</mo><mrow><mo>∫</mo><mrow><mo>∫</mo><mrow><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mrow><mi>ϕ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></mrow></msup><mo>⁢</mo><mrow><msub><mi>had</mi><mi>mn</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mo>-</mo><mrow><mi>ⅈ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><msub><mi>α</mi><mi>mn</mi></msub><mo>⁢</mo><mi>x</mi></mrow><mo>+</mo><mrow><msub><mi>β</mi><mi>mn</mi></msub><mo>⁢</mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></msup><mo>⁢</mo><mrow><mo>ⅆ</mo><mi>x</mi></mrow><mo>⁢</mo><mrow><mo>ⅆ</mo><mi>y</mi></mrow></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mrow><mi>ϕ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></mrow></msup><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover><mo>⁢</mo><mrow><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><msub><mi>ξ</mi><mi>mn</mi></msub></mrow></msup><mo>⁢</mo><mrow><msub><mi>had</mi><mi>mn</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mi>ⅈ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><msub><mi>α</mi><mi>mn</mi></msub><mo>⁢</mo><mi>x</mi></mrow><mo>+</mo><mrow><msub><mi>β</mi><mi>mn</mi></msub><mo>⁢</mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></msup></mrow></mrow></mrow></mrow></mtd></mtr></mtable><mo>}</mo></mrow><mo>⁢</mo><mtable><mtr><mtd><mrow><mi>m</mi><mo>=</mo><mrow><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>…</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>M</mi></mrow><mo>-</mo><mn>1</mn></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mi>n</mi><mo>=</mo><mrow><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>…</mi><mo>⁢</mo><mstyle><mtext> </mtext></mstyle><mo>⁢</mo><mi>N</mi></mrow><mo>-</mo><mn>1</mn></mrow></mrow></mtd></mtr></mtable></mrow></mtd><mtd><mrow><mo>(</mo><mn>18</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> In turn, these equations can be solved iteratively by a Gerchberg-Saxton type algorithm.

00103Kohina, et al. describe a modan designed by this method, whose basis set is the lowest 25 2D-Hadamard functions, <maths id="MATH-US-00023" num="00023"><math overflow="scroll"><mrow><msubsup><mrow><mo>{</mo><mrow><msub><mi>had</mi><mi>mn</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>}</mo></mrow><mrow><mi>mn</mi><mo>,</mo><mrow><mo>=</mo><mn>0</mn></mrow></mrow><mn>4</mn></msubsup><mo>.</mo></mrow></math></maths><br /> See S. N. Kohina, V. V. Kotylar, R. V. Skidanov and V. A. Soifer, “Optical Data Processing Using DOEs” in <i>Methods for Computer Design of Diffractive Optical Elements, </i>V. A. Soifer, ed., John Wiley &amp; Sons, New York, 2002. The entire contents of the Kohina et al. article is hereby incorporated by reference in this application. When followed by a Fourier transforming lens for the purposes of performing an optical Walsh-Hadamard transform, 80% of the incident light falls in the autocorrelation peaks in the focal plane. Therefore this same modan can be used in the optical arrangement of <figref idref="DRAWINGS">FIG. 12A</figref> to make a beam shaper <b>118</b> for a 1×25 switch element <b>10</b> (M=N=5) with about 1 dB of loss.

00105An alternative embodiment, <b>111</b>B, for implementing a (1×K)-switching element is depicted in FIG. <b>10</b>B. In this embodiment, the beam shaping optics <b>116</b> are divided, with part of the optical train preceding the state selector <b>118</b> and part succeeding it.

00106An embodiment <b>111</b>C for implementing an (L×K)-switch (<figref idref="DRAWINGS">FIG. 9A</figref>) directly is depicted in FIG. <b>13</b>. Light from each fiber <b>114</b>A in an array of L input fibers <b>115</b> is collimated by its own micro-lens or GRIN lens <b>112</b>A and directed onto its own the state selector <b>116</b>A. Each state selector <b>116</b>A imposes on its outgoing wavefront the same phase-only wave-amplitude profiles as before. The L state selectors <b>116</b>A can either be individual units, or they can be integrated into a single unit, where different portions of a large SLM are used to spatially modulate the wavefronts from each input fiber. The output of all the state selectors <b>116</b>A then enters a combiner <b>300</b> that projects the exit plane of each state selector <b>116</b>A into the input aperture of the beam shaper <b>118</b>. The combiner is an adaptation of DOEs that have been designed for implementing free-space optical interconnects for computer systems. Beam shaper <b>118</b> can use any of the previously-described DOEs. In the embodiment of <figref idref="DRAWINGS">FIG. 13</figref>, the wave-amplitude on the entrance plane of the beam shaper <b>118</b> is now the linear superposition of wave-amplitudes from each of the L state selectors: <maths id="MATH-US-00024" num="00024"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mi>u</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>L</mi></munderover><mo>⁢</mo><mrow><mi>u</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mrow><mi>y</mi><mo>;</mo><mrow><mi>n</mi><mo>⁡</mo><mrow><mo>[</mo><mi>l</mi><mo>]</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>19</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where u(x,y;n) is the wave-amplitude generated at the input aperture of the beam shaper by a state selector when that state selector is set to direct its light to output fiber n; n[l] is the state that state selector l is set to. The beam shaper functions as before. Since it is a linear device, it performs the same transformation on each of the component wave-amplitudes u(x,y;n n[l]) in equation (19). Therefore the light from input fiber l is ultimately focused onto the collection lens/output optical fiber n[l], <b>119</b> and <b>120</b>.

00108While the invention has been described in connection with what is presently considered to be the most practical and preferred embodiment, it is to be understood that the invention is not to be limited to the disclosed embodiment, but, on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.